13 MEASURES WITH BASE // 181 this linear form is a (complex) measure on X, because if K is any compact subset of X, we have i r fe I \\f\\ for all/e Jfc(X; K), by virtue of (13.10.3). The measure so defined is called the measure with density g relative to /*, and is denoted by g • ju; when g is continuous, this agrees with the definition (13.1.5). Measures of the form g - n are also called measures with base //. It follows immediately from this definition that if g takes values in R (resp. ^0 almost everywhere), then the measure g • u is real (resp. positive). Furthermore, g • fi does not change if we replace g by an equivalent function (with respect to u), and therefore we may restrict ourselves to the case where g is everywhere finite and universally measurable (13.9.12). If g± and g2 are two locally integrable functions, then g^ + g2 and agt are locally integrable (a being any complex scalar), and we have (13.13.2) fa + ff2) • \i = #! -11 + g2 ' n, (agj - fi For every complex-valued locally integrable function g, we have (13.13.3) # • /* = £ • //, »(g-fi = (&g) ' I*, Sfa - p) * (Sg) (13.13.4) The set &\^ R(X, u) (resp. &\^ C(X, JM)) of real-valued (resp. complex-valued) locally ^-integrable functions is a real (resp. complex) vector space, often denoted by J?loc(X, /*) or J?IOO(A«) or JS? loc(X) or ^,oc , For every compact subset K of X, the mapping pK : g\-+ f |^K| rf/i is a semi-norm on this space. We shall always suppose jSf ,oc to be endowed with the topology defined by these seminorms. If (Kw) is an increasing sequence of compact subsets of X, such that (J Kn = X and Kn c Kn + 1 (3.18.3), then it is n immediately seen that the topology of & ,oc is defined by the seminorms pKn. In particular, if X is compact, then jS?/OCt R(X, fi) (resp. j?,OCp C(X, ju)) is identical with JS?i(X, //) (resp. ^c(x» j"))- The 'set of locally integrable real- valued functions g such that pK(g) = 0 for all compact subsets K of X is the space Jf of ^-negligible functions. We define and L/OCj C(X, p) analogously. The seminorms p^(g) depend only on the class g of g: if we put pK(g) = p^g), then we obtain seminorms defining the topologies of these spaces, which are therefore metrizable and locally convex. a locally ^-integrable function. Since /H* \fg d^ is defined on the